Property ( w) and perturbations II

Abstract

This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property ( w ) and perturbations, J. Math. Anal. Appl. 336 (2007) 683–692] concerning the stability of property ( w ) , a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T. A counterexample shows that property ( w ) in general is not preserved under finite-dimensional perturbations commuting with T, also under the assumption that T is a-isoloid.